Necessary adiabatic run times in quantum optimization

Lucas T. Brady and Wim van Dam

Phys. Rev. A 95, 032335 – Published 30 March 2017

Abstract

Quantum annealing is guaranteed to find the ground state of optimization problems provided it operates in the adiabatic limit. Recent work [S. Muthukrishnan et al., Phys. Rev. X 6, 031010 (2016)] has found that for some barrier tunneling problems, quantum annealing can be run much faster than is adiabatically required. Specifically, an $n$ -qubit optimization problem was presented for which a nonadiabatic, or diabatic, annealing algorithm requires only a constant run time, while an adiabatic annealing algorithm requires a run-time polynomial in $n$ . Here we show that this nonadiabatic speedup is the direct result of a specific symmetry in the studied problem. In the more general case, no such nonadiabatic speedup occurs and we show why the special case achieves this speedup compared to the general case. We also prove that the adiabatic annealing algorithm has a necessary and sufficient run time that is quadratically better than the standard quantum adiabatic condition suggests. We conclude with an observation about the required precision in timing for the diabatic algorithm.

Received 6 February 2017

DOI:https://doi.org/10.1103/PhysRevA.95.032335

Physics Subject Headings (PhySH)

Quantum algorithms Quantum computation

Quantum Information, Science & Technology

Authors & Affiliations

Lucas T. Brady^*

Department of Physics, University of California, Santa Barbara, California 93106-5110, USA

Wim van Dam

Department of Computer Science, Department of Physics, University of California, Santa Barbara, California 93106-5110, USA

^*Corresponding author: lbrady@physics.ucsb.edu

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Vol. 95, Iss. 3 — March 2017

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Images

Figure 1
The single qubit success probability, $p_{1}$ , as a function of the total run time for several $μ$ values. The dashed blue, $μ = 1$ line corresponds to the model that has been studied in previous articles. Note that the $μ = 1$ curve has several special properties, including that it goes to $p_{1} = 1$ at finite $τ$ , resulting in the nonadiabatic speedup noted in other papers. The $μ \neq 1$ curves do not exhibit this $p_{1} = 1$ behavior.
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Figure 2
Optimal expected running time of quantum annealing, $τ_{n} / p_{n} (τ_{n})$ , as a function of $n$ for different $μ$ values. Unlike the $μ = 1$ case, $τ_{n}$ increases with $n$ for these $μ$ values. Lines through the symbols are power-law fits of the form $τ_{n} = A n^{r}$ , and the fitted $r$ values in the order $μ = (0.6, 0.8, 1.2, 1.4, 1.6)$ are $(0.48, 0.51, 0.49, 0.48, 0.46)$ , all close to $1 / 2$ . A scaling power of $1 / 2$ is consistent with the adiabatic scaling of the $μ = 1$ case as found in [2] and our results below while being quadratically faster than the sufficient adiabatic condition.
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Figure 3
A single-qubit success probability curve as a function of the total run time $τ$ for $μ = 1.5$ , with upper and lower envelopes shown. These envelopes were obtained by first extracting all the local minima (maxima) and doing a power-law fit of the form $p_{1} = 1 - c τ^{- q}$ . The first two minima and maxima were excluded from this fit and others since they tend to be more abnormal. In this case, the upper envelope has a fitted $q = 1.998$ and the lower envelope has a fitted $q = 1.996$ , both of which are extremely close to the 2 we use in the text.
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Figure 4
Curves like the one in Fig. 3 are bounded above and below by curves of the form $1 - c / τ^{2}$ . We show the values of $c$ for the upper, $c_{u}$ , and lower, $c_{ℓ}$ , bounding functions as obtained from numerical fits. These coefficients are a function of $μ$ , and all of the fits used to obtain this data were good quality. In the text, we show that these bounding curves directly lead to a $O (\sqrt{n})$ running time for the algorithm in all cases except the $μ = 1$ case, where $c_{u} (1) = 0$ .
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Figure 5
This plot shows the run time, $τ$ , needed to ensure that the state of the system is at least $75 %$ in the ground state over the entire $s$ evolution. This growth of $τ$ with $n$ comes closest to a true adiabatic evolution, and we can see that the $τ \in O (\sqrt{n})$ behavior holds even in this case. Power-law fits to these data sets show that the exponent for these curves, in the order $μ = (0.5, 1.0, 1.5, 2.0)$ , are $(0.497, 0.502, 0.501, 0.500)$ . Therefore, the quadratic speedup we see over the sufficient adiabatic condition is a property of adiabatic evolution in this system, not the specific $τ / p_{n}$ criteria we used.
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Figure 6
The width of the nonadiabatic success probability peak in the curve of $p_{n}$ vs $τ$ is shown as a function of $n$ . This problem tunnels through a binomial barrier of width and height $\propto n^{0.3}$ . The width of the peaks in success probability are extracted by performing a Gaussian fit to the top of the peak and extracting the width of the fitted Gaussian. The width is modeled well by the curve $w = A n^{- p}$ , where $A = 13.97$ and $p = 0.467$ . This width scaling with $n$ is close to the width scaling of $n^{- 1 / 2}$ extracted analytically for the no-barrier case.
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